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Series: High Dimensional Seminar

TBA

Series: High Dimensional Seminar

We discuss the asymptotic value of the maximal perimeter of a convex set in an n-dimensional space with respect to certain classes of measures. Firstly, we derive a lower bound for this quantity for an arbitrary probability measure with first two moments bounded; the lower bound depends on the moments only. This lower bound is sharp in the case of the Gaussian measure (as was shown by Nazarov in 2001), and, more generally, in the case of rotation invariant log-concave measures (as was shown by the author in 2014). We show, that this lower bound is also sharp for a class of smooth log-concave measures satisfying certain uniform bounds on the hessian of the potential. In addition, we show a uniform upper bound of Cn for all isotropic log-concave measures, which is attained for the uniform distribution on the cube. Some improved bounds are also obtained for the Poisson density.

Series: PDE Seminar

Monday, April 29, 2019 - 13:55 ,
Location: Skiles 005 ,
Prof. Siu A. Chin ,
Texas A&M University ,
chin@physics.tamu.edu ,
Organizer: Molei Tao

TBA

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Given an m-dimensional manifold M that is homotopy equivalent to an n-dimensional manifold N (where n<m), a spine of M is a piecewise-linear embedding of N into M (not necessarily locally flat) realizing the homotopy equivalence. When m-n=2 and m>4, Cappell and Shaneson showed that if M is simply-connected or if m is odd, then it contains a spine. In contrast, I will show that there exist smooth, compact, simply-connected 4-manifolds which are homotopy equivalent to the 2-sphere but do not contain a spine (joint work with Tye Lidman). I will also discuss some related results about PL concordance of knots in homology spheres (joint with Lidman and Jen Hom).

Series: Algebra Seminar

TBA

Series: CDSNS Colloquium

Friday, April 19, 2019 - 14:00 ,
Location: Skiles 006 ,
Arash Yavari and Fabio Sozio, School of Civil and Environmental Engineering ,
Georgia Tech ,
Organizer: Igor Belegradek

We formulate a geometric nonlinear theory of the mechanics of accretion. In this theory the material manifold of an accreting body is represented by a time-dependent Riemannian manifold with a time-independent metric that at each point depends on the state of deformation at that point at its time of attachment to the body, and on the way the new material isadded to the body. We study the incompatibilities induced by accretion through the analysis of the material metric and its curvature in relation to the foliated structure of the accreted body. Balance laws are discussed and the initial-boundary value problem of accretion is formulated. The particular cases where the growth surface is either fixed or traction-free are studied and some analytical results are provided. We numerically solve several accretion problems and calculate the residual stresses in nonlinear elastic bodies induced from accretion.

Series: Stochastics Seminar